Not only AM-GM 4 (The Other AM-GM Part 6)

Algebra Level 5

Positive real numbers $a$ , $b$ , $c$ , and $d$ are such that $a = 27 - c$ and $\dfrac{1}{b} = 21 - \dfrac{1}{d}$ . Find the maximum value of the expression below to 3 decimal places.

$\large\dfrac{1}{ \dfrac{1}{a} + b + \dfrac{1}{c} + d}$

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The answer is 2.953.

1 solution

Anirudh Sreekumar
Apr 24, 2017

$\begin{aligned} a+c&=27 \hspace{7mm}\small\color{#3D99F6} (\text{Given})\\ \dfrac1 b+\dfrac 1 d&=21\hspace{7mm}\small\color{#3D99F6} (\text{Given}) \\ \text{Applying AM}&\geq\text{HM, We have,}\\\\ \dfrac{a+c}{2}&\geq\dfrac{2}{\dfrac1 a+\dfrac 1 c}\\ \implies \dfrac1 a+\dfrac 1 c&\geq\dfrac{4}{a+c}\\ &\geq\dfrac4 {27}\\ \text{Similarly,}\\\\ \dfrac{\dfrac1 b+\dfrac 1 d}{2}&\geq\dfrac{2}{b+d}\\ \implies b+d&\geq\dfrac{4}{\dfrac1 b+\dfrac 1 d}\\ &\geq\dfrac4 {21}\\\\ \implies \dfrac1 a +b+\dfrac1 c +d&\geq\dfrac4 {27}+\dfrac4 {21}\\\\ \implies\dfrac{1}{ \dfrac1 a +b+\dfrac1 c +d}&\leq\dfrac{27\times 21}{4\times48}\\ &\leq2.953125\end{aligned}$

Not only AM-GM 4 (The Other AM-GM Part 6)

Try another problem on my set Let's Practice

The answer is 2.953.

1 solution

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